Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Which of the following is (are) not the square of a $3 \times 3$ matrix with real entries:
$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$
$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$
$\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$
$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
Step-by-Step Solution
Key Concept: If A = B² for real matrix B, then det(A) = det(B²) = (det(B))² ≥ 0. A matrix can only be a perfect square if its determinant is non-negative.
Given $A = B^2$, we have $|A| = |B^2| = |B|^2 \geq 0$, always non-negative. For (A): $|A| = |-1| = -1$ (negative), impossible. For (B): $|A| = |1(−0) − 0(0)| = 1$ (positive), possible. For (C): Example computation yields determinant $−1$ (negative), impossible. For (D): Determinant equals $1$ (positive), possible.
Correct Answer: 1,3